\(\int \frac {\cos (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx\) [224]

Optimal result

Integrand size = 24, antiderivative size = 12 \[ \int \frac {\cos (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=-\frac {\log (\cos (c+d x))}{d} \]

[Out]

-ln(cos(d*x+c))/d

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {4419, 266} \[ \int \frac {\cos (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=-\frac {\log (\cos (c+d x))}{d} \]

[In]

Int[Cos[c + d*x]/(Csc[c + d*x] - Sin[c + d*x]),x]

[Out]

-(Log[Cos[c + d*x]]/d)

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 4419

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, Dist[d/(b
*c), Subst[Int[SubstFor[1, Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d], x] /; FunctionOfQ[Sin[c*(a +
b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Cos] || EqQ[F, cos])

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d} \\ & = -\frac {\log (\cos (c+d x))}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=-\frac {\log (\cos (c+d x))}{d} \]

[In]

Integrate[Cos[c + d*x]/(Csc[c + d*x] - Sin[c + d*x]),x]

[Out]

-(Log[Cos[c + d*x]]/d)

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.08

method	result	size
derivativedivides	\(-\frac {\ln \left (\cos \left (d x +c \right )\right )}{d}\)	\(13\)
default	\(-\frac {\ln \left (\cos \left (d x +c \right )\right )}{d}\)	\(13\)
risch	\(i x +\frac {2 i c}{d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\)	\(30\)
parallelrisch	\(\frac {-\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{d}\)	\(46\)
norman	\(\frac {\ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{d}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\)	\(54\)

[In]

int(cos(d*x+c)/(csc(d*x+c)-sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-ln(cos(d*x+c))/d

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cos (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=-\frac {\log \left (-\cos \left (d x + c\right )\right )}{d} \]

[In]

integrate(cos(d*x+c)/(csc(d*x+c)-sin(d*x+c)),x, algorithm="fricas")

[Out]

-log(-cos(d*x + c))/d

Sympy [F]

\[ \int \frac {\cos (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=\int \frac {\cos {\left (c + d x \right )}}{- \sin {\left (c + d x \right )} + \csc {\left (c + d x \right )}}\, dx \]

[In]

integrate(cos(d*x+c)/(csc(d*x+c)-sin(d*x+c)),x)

[Out]

Integral(cos(c + d*x)/(-sin(c + d*x) + csc(c + d*x)), x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 24, normalized size of antiderivative = 2.00 \[ \int \frac {\cos (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=-\frac {\log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )}{2 \, d} \]

[In]

integrate(cos(d*x+c)/(csc(d*x+c)-sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/2*(log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1))/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (12) = 24\).

Time = 0.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 2.17 \[ \int \frac {\cos (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=-\frac {\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) + \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{2 \, d} \]

[In]

integrate(cos(d*x+c)/(csc(d*x+c)-sin(d*x+c)),x, algorithm="giac")

[Out]

-1/2*(log(abs(sin(d*x + c) + 1)) + log(abs(sin(d*x + c) - 1)))/d

Mupad [B] (verification not implemented)

Time = 22.69 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cos (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=-\frac {\ln \left ({\cos \left (c+d\,x\right )}^2\right )}{2\,d} \]

[In]

int(-cos(c + d*x)/(sin(c + d*x) - 1/sin(c + d*x)),x)

[Out]

-log(cos(c + d*x)^2)/(2*d)